1. ICCBT2008
A Serviceability Approach to the Design of SCC beams
H. Narendra, M S Ramaiah Institute of Technology, INDIA
K. U. Muthu*, M S Ramaiah Institute of Technology, INDIA
H. M. A. Al-Mattarneh, Universiti Tenaga Nasional, MALAYSIA
S. Ganesh, M S Ramaiah Institute of Technology, INDIA
M. Vijayanand, M S Ramaiah Institute of Technology, INDIA
ABSTRACT
The slender structural elements are preferred nowadays from aesthetic point of view. This
necessitates the control of deflections. Various international codes of practices specify span to
effective depth ratios to control the deflections of conventional concrete beams. A basic span
to effective depth ratio is specified and suitably modified depending the tensile ratio, grade of
steel, compression steel ratio and the flange widths. A similar specification for SCC beams
has not been come across as the modulus of elasticity and modulus of rupture of SCC vary
considerably with respect to the similar properties of conventional concrete. A method is
proposed to arrive at the span to effective ratio of SCC beams. The method has been
developed using a simplified effective moment of inertia function proposed by the authors
recently. The method of computation has been verified with the test data. Design charts are
presented for ready use. The effective depths of beams with and without compression
reinforcement were obtained using the proposed method and compared with those obtained
from limit state of strength and minimum depth specified by the code. The application of the
method is illustrated with examples.
Keywords: Beams, Chart, Code, Comparison, Concrete, Effective moment of inertia, Self
Compacting,Tests.,
Correspondence Author: Pro.Dr.K.U.Muthu, M S Ramaiah Institute of Technology, India, Tel:+919845363314,
Fax:+918023603124,E-mail: kumuthu@rediffmail.com
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2. A Serviceability Approach to the Design of SCC beams
1. INTRODUCTION:
ACI3181, BS81102 and IS4563 recommend two approaches for the serviceability
criteria – deflection. In the first approach i.e, computational approach – the total deflection
was computed using the elastic deflection formula. The computed deflections were then
compared with the permissible deflection expressed as a function of span length. In the second
approach, i.e. control approach – the span to effective depth ratios or minimum thickness as a
function of span length is specified. It is implied that if the beams / slabs are designed as per
the above specification, the resulting total deflection under working loads is within the
permissible limits. Several investigators pointed out the disparities and discrepancies of the
above approach4. A few investigations were reported considering the sustained load effect.5-8.
An attempt has been made in the present study, to propose a limiting span to effective depth
ratios of SCC beams.
The method is based on the effective moment of inertia suggested in Eurocode9 and a similar
simplified alternate form was proposed by author’s elsewhere10. The resulting effective depths
have been compared with those obtained from rupture limit state and deflection limit state.
2. PROPOSED METHOD
2.1 Short-term deflection
The central deflection of a simply supported beam subjected to uniformly distributed load is
given as
αwl 4
δ inc = (1)
Ec I eff
where α =deflection coefficient depending on support conditions viz cantilever, one end
continuous, both end continuous and simply supported. As the beam considered is simply
supported beam, it is taken as 5/384; w is the udl per unit length, l is the span of the beam,
E c =modulus of elasticity of self compacting concrete taken as 3750 f ck where f ck is the
cube compressive strength in MPa; I eff is the effective moment of inertia of the section.
2.2 Effective Moment of Inertia Function: I eff
The CEB-FIP Model code 19909 recommends the effective moment of inertia as
I g I cr
Ie = (2)
⎡ ⎛ M cr ⎞ ⎤
2
⎜ M ⎟ ⎥ (I g − I cr )
I cr + ⎢1 − 0.5⎜ ⎟
⎢
⎣ ⎝ a⎠ ⎥ ⎦
where I g , I cr and I eff are the gross, cracked and effective moment of inertia, M cr and M a are
the cracking moment and actual moment respectively. This equation is simplified in the form
as
I e = kbd 3 (3)
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3. H. Narendra et al.
k = 27.05ρ + 0.338 (4)
ρ is the steel ratios and the correlation coefficient as 0.85 (Fig.1)
2
1.8
1.6
1.4
1.2
I eff/(bd3/12)
1
0.8
0.6
0.4
0.2
0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
% of steel
Figure 1. Variation of Percentage of Steel Vs Ieff
⎛l⎞
2.3 Limiting Span to Effective Depth Ratio: ⎜ ⎟
⎝d ⎠
The total deflection is obtained from short term and additional long term deflection as
αwl 4 λαwS l 4
Δ= + (5)
Ec I eff Ec I eff
where λ is the long time multiplier for sustained loads. Taking the target deflection as
span/250 and using Eq.(3);
l αwl 4 λαwS l 4
= + (6)
250 Ec kbd 3 Ec kbd 3
The equation is simplified in the following form to obtain the span to effective ratios as
1
l ⎡ Ec kb ⎤3
=⎢ ⎥ (7)
d ⎣ 250αw(1 + λws / w) ⎦
The above equation is plotted and shown in (Fig 2) for various values of sustained load to the
service load ratios and steel ratios for singly reinforced beams.
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4. A Serviceability Approach to the Design of SCC beams
0.29
0.27
ρ = 0.005
[l/d*(w/(b*Ec))](1/3 )
0.25
ρ = 0.01
0.23 ρ = 0.015
0.21 ρ = 0.02
ρ = 0.025
0.19
ρ = 0.03
0.17
0.15
0 0.2 0.4 0.6 0.8 1 1.2
w s/w
1
l ⎛ w ⎞ 3
Figure 2. Variation of ⎜ ⎟ Vs (ws/w)
d ⎜ bEc ⎟
⎝ ⎠
2.4 (l d ) Ratios for Beams with Compression Reinforcement.
The procedure for obtaining (l d ) ratios has been extended to beams with compression
reinforcement. The method of obtaining the simplified effective moment of inertia has been
reported in reference [10] by using the appropriate use of modulus of elasticity and modulus
of rupture of self compacting concrete. The modulus of rupture of SCC fr, was taken as
2
0.4( f ck )3 MPa. The simplified equation is of the form of Eq(3)
⎛ ρ + ρ' ⎞
⎜ ρ ⎟ + 0.0734
k = 0.014⎜ ⎟ ...(8)
⎝ b ⎠
and the correlation coefficient of the above equation was found to be 0.74, (Fig3)
0.16
0.14
0.12
3
Ieff /bd
0.1
0.08
0.06
0.04
1 1.5 2 2.5 3 3.5 4 4.5 5
(ρ+ρ')/(ρb)
Figure 3. Variation of Ieff Vs Steel Ratio for all Grades of Steel
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5. H. Narendra et al.
The span to effective depth ratio for beams with compression reinforcement has been obtained
using Eq(7). It is to be noted that the value of k corresponds to Eq(8). A graphical
representation of Eq(7) is presented in (Fig 2) for ready use.
0.14
0.13
0.12
[l/d*(w/(b*Ec))](1/3)
(ρ+ρ')/(ρb) = 1.03
(ρ+ρ')/(ρb) = 1.75
0.11 (ρ+ρ')/(ρb) = 2.5
(ρ+ρ')/(ρb) = 3.25
(ρ+ρ')/(ρb) = 4
0.1
(ρ+ρ')/(ρb) = 4.76
0.09
0.08
0 0.2 0.4 0.6 0.8 1 1.2
w s/w
1
l ⎛ w ⎞3
Figure 4. Variation of ⎜ ⎟ Vs ws/w
d ⎜ bEc ⎟
⎝ ⎠
Example 1:
Compute the effective depth of a singly reinforced beam for the following data, Breadth of the
beam=230mm, Span of the beam=6m, service load=25kN/m, Grade of concrete=60MPa and
Grade of steel=415MPa. Obtain the effective depths from the limit state of strength and limit
state of deflection as per ACI3181.
1
l ⎛ w ⎞3
Using Fig2; for a known value of (ws/w) and the steel ratio (ρ), the value of ⎜ ⎟ was
d ⎜ bEc ⎟
⎝ ⎠
obtained. The value of‘d’ depth of the beam was obtained by substituting the value of w, b
and Ec. Also the effective depth of the beam was obtained using limit state of strength and
deflection as per ACI3181.
Table 1. Comparison of Effective Depths of Singly Reinforced Beams
ρ d(mm) limit state of 0.2 0.4 0.6 0.8 1.0
Strength deflection
0.50 892.13 375 453.77 493.43 527.56 557.77 585.02
0.75 634.95 375 434.01 471.93 504.58 533.47 559.54
1.00 521.86 375 417.29 453.75 485.14 512.92 537.98
1.25 454.97 375 402.88 438.09 468.39 495.22 519.41
1.50 409.70 375 390.28 424.38 453.74 479.72 503.16
1.75 376.58 375 379.12 412.25 440.76 466.00 488.77
2.00 351.07 375 369.13 401.39 429.16 453.73 475.90
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6. A Serviceability Approach to the Design of SCC beams
It is noted from Table 1, that when the steel ratio and the ratio of sustained load to service
load is less, the effective depth obtained from limit state of strength is found to be greater than
the effective depths obtained from limit state of deflection. When the steel ratio and ratio of
sustained load to service load is larger, the effective depths obtained from limit state of
strength is to be increased to a higher value.
Example 2:
Estimate the effective depth of a doubly reinforced rectangular beam for the following data,
Breadth of the beam=230mm, Span=8m, Service load=30kN/m, Grade of concrete = 30MPa,
Steel grade=415MPa, Compression and Tensile steel ratios are listed in Table 2.
The same procedure for singly reinforced beams has been extended to doubly reinforced
beams using Fig (4). The effective depths thus obtained are given in Table 2 with
1
those obtained from limit state of strength and deflection as per ACI318 .
Table 2. Comparison of Effective Depths of Doubly Reinforced Beams
ρ ρ1 ρ + ρ1 ACI-318 Proposed method for ws/w
Strength Deflection 0.2 0.4 0.6 0.8 1
limit limit
1.885 0.478 2.363 432.75 500 534.68 581.40 621.62 657.22 689.33
2.036 0.642 2.678 417.97 500 528.57 574.76 614.52 649.71 681.45
2.190 0.806 2.996 404.69 500 522.68 568.35 607.67 642.47 673.85
2.344 0.97 3.314 392.71 500 516.96 562.14 601.02 635.44 666.49
2.498 1.134 3.632 381.82 500 511.41 556.10 594.57 628.62 659.33
2.652 1.299 3.951 371.82 500 505.95 550.16 588.22 621.91 652.29
It is noted from Table 2, that the effective depths obtained from limit state of deflection of
ACI 3181 larger than these obtained from limit state of strength. However the proposed
method indicates that they are inadequate to satisfy the target total deflection of span/250.
3. SUMMARY AND CONCLUSIONS
A design method is proposed to determine the effective depths of self-compacting concrete
beams from serviceability limit state of deflection. The proposed method is based on the
effective moment of inertia function given in CEB-FIP code. The results of the proposed
method indicates (i) when steel ratio is less and for different sustained to service load ratios,
the effective depths obtained from limit state of strength governs the design of singly
reinforced beams while for large steel ratios and higher value of (ws/w), the effective depths
obtained by the codal provisions are inadequate. (ii) in case of beams with compression
reinforcement; the effective depths obtained from the codal provisions are to be increased to a
higher value to satisfy the target total deflection of span/250.
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7. H. Narendra et al.
Acknowledgement
The authors would like to acknowledge the management of M.S.Ramaiah Institute of
Technology, Bangalore and University Tenaga Nasional, Malayasia, the Principal Dr. K.
Rajanikanth of M.S.Ramaiah Institute of Technology, and the higher authorities of UNITEN,
Malaysia for their constant encouragement throughout the course of study.
Notation
b = breadth of the beam
d = effective depth of the beam
Ec = modulus of elasticity of self compacting concrete; 3750 f ck in MPa
f ck = cube compressive strength of concrete in MPa
2
fr = modulus of rupture, 0.4 f ck 3 in MPa
I cr = cracked moment of inertia of the section
Ig = gross moment of inertia of the section neglecting reinforcement
Ie = effective moment of inertia of the section
l = span of the beam
w = service load/m
δ = short term deflection
Δ = total deflection of the beam
λ = additional long time multiplier
ρ = tensile steel ratio
ρ1
= compression steel ratio
ρb = balanced steel ratio
ρt = sum of compression and tensile ratio
α = deflection coefficient
M cr = cracking moment
Ma = actual moment
REFERENCES
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and commentary (318R-05), American Concrete Institute, Farmington Hills, Mich, 430pp.
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Standards, Fourth Revision, New Delhi.
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prestressed concrete beams and slabs”, ACI Structural Journal, V103, No1, Jan-Feb 2006, pp
142-148.
[5]. Rangan, B.V, (1982), “Control of Beam deflections by Allowable Span –to-depth Ratios”, ACI
Journal , Proceedings, V-79, No.5, Sep 1982, pp 372-377.
ICCBT 2008 - C - (16) - pp183-190 189

8. A Serviceability Approach to the Design of SCC beams
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